Dear Folks,

Calculate the straight line characteristics for the following linear 1st order PDE's

1) $\displaystyle 3u_x +5u_y -xyu=0$

I calculate $\displaystyle k=y-\frac{5}{3}x$ and when transformed becomes $\displaystyle \eta=y-\frac{5}{3}\xi$

but the book its given as $\displaystyle 5x-3y=k$ and when transformed becomes $\displaystyle 5\xi-3y=\eta$

2) $\displaystyle yu_x+x^2uy=xy$

I calculate $\displaystyle k=-\frac{x^3}{3}+\frac{y^2}{2}$ when transformed becomes $\displaystyle \eta=\frac{-\xi^3}{3}+\frac{y^2}2}$

but the book is $\displaystyle k=2x^3-3y^2$ when transformed becomes $\displaystyle \eta=2\xi^3-3y^2$

The only way I can see them arriving at this is $\displaystyle k'=\frac{-x^3}{3}+\frac{y^2}{2}$ therefore

$\displaystyle -6k'=2x^3-3y^2=k$

The pattern I see in these kind of questions is that when k is written in terms of x and y that the 'x' terms are always positve before transforming. Why is this?

Thanks